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arxiv: 2606.26174 · v1 · pith:2FQVUAAZnew · submitted 2026-06-24 · 🌀 gr-qc · astro-ph.HE· hep-th

Electrically Charged Distorted Black Holes: Thermodynamics, Particle Dynamics, and Quasinormal Signatures

Gamal G.L. Nashed , Salvatore Capozziello This is my paper

Pith reviewed 2026-06-26 01:38 UTC · model grok-4.3

classification 🌀 gr-qc astro-ph.HEhep-th
keywords distorted black holesEinstein-Maxwell theoryHarrison transformationblack hole thermodynamicsquasinormal modesblack hole shadowparticle dynamicssuperradiance
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The pith

The Killing horizon of an electrically charged distorted black hole is fixed solely by the seed metric.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper applies the Harrison transformation to generate an exact electrically charged version of a distorted black hole in Einstein-Maxwell theory. The key result is that the location of the Killing horizon stays exactly where it was in the uncharged seed geometry, unaffected by the added charge. Thermodynamic quantities such as entropy and temperature follow from the geometry alone, while the charge adds a new thermodynamic variable. The distortion parameter alters the motion of charged particles by shifting the innermost stable orbit and enlarges the black hole shadow by moving the photon sphere outward. Charged scalar fields do not exhibit superradiant instability because the electric potential vanishes on the horizon.

Core claim

We construct an exact solution for the electrically charged extension of a distorted black hole spacetime within Einstein-Maxwell theory using the Harrison transformation. The resulting solution represents a charged deformation of a static distorted vacuum geometry in which the electromagnetic field is introduced through a nonlinear transformation preserving the radial structure of the seed spacetime. Consequently, the Killing horizon remains determined solely by the seed metric and it is not shifted by the electric charge.

What carries the argument

Harrison transformation applied to a static distorted vacuum seed metric, preserving radial structure while adding the electromagnetic field.

If this is right

  • The horizon area, entropy, and temperature are governed by the geometric sector of the solution.
  • The electric charge enlarges the thermodynamic phase space through the electromagnetic potential.
  • The distortion parameter modifies circular orbits of charged test particles and shifts the location of the innermost stable circular orbit.
  • The distortion parameter displaces the photon sphere outward, increasing the apparent shadow size for a static observer.
  • The vanishing horizon electric potential prevents charged superradiant amplification in scalar perturbations.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The method may extend to other vacuum seed metrics with preserved radial structure for generating new charged solutions.
  • The geometric correspondence between the photon orbit and eikonal quasinormal modes could link electromagnetic and gravitational wave observations.
  • Absence of superradiance suggests these geometries remain stable against charged scalar field instabilities.
  • The WKB approximation for quasinormal frequencies offers a benchmark for full numerical evolution of perturbations.

Load-bearing premise

There exists a static distorted vacuum seed metric to which the Harrison transformation applies while preserving the radial structure and producing a valid Einstein-Maxwell solution.

What would settle it

Direct evaluation of the norm of the timelike Killing vector in the charged metric, confirming it vanishes at the identical radial coordinate as the seed metric regardless of the value of the electric charge parameter.

Figures

Figures reproduced from arXiv: 2606.26174 by Gamal G.L. Nashed, Salvatore Capozziello.

Figure 1
Figure 1. Figure 1: Illustration of particle dynamics and scalar perturbations in the charged distorted black hole spacetime. Panel (a) shows the dependence of the innermost stable circular orbit (ISCO) radius on the distortion parameter B, demonstrating that the location of the marginally stable orbit shifts as the external distortion increases. Panel (b) displays the effective potential for different values of B, indicating… view at source ↗
Figure 2
Figure 2. Figure 2: Shadow properties of the charged distorted black hole. Panel (a) shows the shadow contour in the observer sky for several values of the distortion parameter B. The curves correspond to B = 0.0, B = 0.1, B = 0.2, and B = 0.3. As the distortion parameter increases, the shadow radius becomes slightly larger while preserving an approximately circular shape, indicating that the external distortion modifies the … view at source ↗
Figure 3
Figure 3. Figure 3: Illustration of scalar perturbations in the charged distorted black hole spacetime. (a) Scalar effective potential Vsc(r) evaluated on the equatorial plane for several values of the scalar charge qs. The potential exhibits a single-barrier structure outside the event horizon, which is the typical condition required for the WKB approximation used in quasinormal-mode calculations. (b) Gauge-induced frequency… view at source ↗
read the original abstract

We construct an exact solution for the electrically charged extension of a distorted black hole spacetime within Einstein-Maxwell theory using the Harrison transformation. The resulting solution represents a charged deformation of a static distorted vacuum geometry in which the electromagnetic field is introduced through a nonlinear transformation preserving the radial structure of the seed spacetime. Consequently, the Killing horizon remains determined solely by the seed metric and it is not shifted by the electric charge. We analyze the thermodynamic properties of the solution and show that the horizon area, entropy, and temperature are governed by the geometric sector, while the electric charge enlarges the thermodynamic phase space through the electromagnetic potential. The motion of charged test particles is studied using the effective potential formalism, where the distortion parameter modifies circular orbits and shifts the location of the innermost stable circular orbit. We also investigate the black hole shadow for a static observer at finite distance and show that the distortion parameter displaces the photon sphere outward, increasing the apparent shadow size. A geometric correspondence between the photon orbit, determining the shadow and the leading eikonal quasinormal-mode frequency, is discussed, linking optical and perturbative observables. Finally, we study charged scalar perturbations and show that the vanishing horizon electric potential prevents a charged superradiant amplification. In the weak-coupling regime, the quasinormal-mode spectrum is estimated using the WKB method, where the electromagnetic interaction enters through the gauge-invariant combination $(\omega - q_s \xi_t)$ and shifts the oscillation frequencies of the perturbations.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper constructs an exact electrically charged distorted black hole solution in Einstein-Maxwell theory by applying the Harrison transformation to a static distorted vacuum seed metric. It claims that the transformation preserves the radial structure so that the Killing horizon location is fixed solely by the seed and independent of charge. The work then analyzes thermodynamics (area, entropy, temperature governed by geometry; charge enters only via potential), charged test-particle dynamics (effective potential, ISCO shifts due to distortion), the shadow for a static observer (distortion displaces photon sphere outward), a geometric link between photon orbits and eikonal QNMs, and charged scalar perturbations (no superradiance because horizon potential vanishes; WKB frequencies shifted by gauge-invariant combination).

Significance. If the construction and horizon non-shift claim are valid, the solution supplies a controlled family of charged distorted black holes in which charge enlarges the thermodynamic phase space without altering the horizon geometry, enabling clean separation of geometric and electromagnetic effects in particle orbits, shadows, and perturbations. The explicit link between photon-sphere radius and leading eikonal QNM frequency is a useful observable correspondence.

major comments (2)
  1. [Abstract / construction] Abstract and construction section: the claim that the Killing horizon remains determined solely by the seed metric and is not shifted by electric charge is load-bearing for the entire solution class. When the distortion parameter is set to zero the seed reduces to Schwarzschild; the standard Harrison transformation on Schwarzschild produces the Reissner-Nordström metric whose outer horizon radius explicitly depends on Q. The manuscript must exhibit the explicit metric components (or at least the norm of the static Killing vector) in this limit and demonstrate either that the generated line element differs from RN or that a non-standard form of the transformation is being used that still satisfies the Einstein-Maxwell equations.
  2. [Thermodynamics] Thermodynamics section: if the horizon radius is truly independent of Q, the first law and Smarr relation must be re-derived with the electromagnetic potential appearing only as an external parameter; the manuscript should show the explicit form of these relations and verify consistency with the area law when the seed is Schwarzschild.
minor comments (2)
  1. Notation for the distortion parameter and the electromagnetic potential should be introduced with a single consistent symbol and its range stated explicitly.
  2. [Quasinormal modes] The WKB formula for the QNM frequencies should include the precise order of the approximation and the range of multipoles for which it is applied.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for identifying key points that require clarification in the construction and thermodynamic analysis. We respond to each major comment below and will implement the indicated revisions.

read point-by-point responses

  1. Referee: [Abstract / construction] Abstract and construction section: the claim that the Killing horizon remains determined solely by the seed metric and is not shifted by electric charge is load-bearing for the entire solution class. When the distortion parameter is set to zero the seed reduces to Schwarzschild; the standard Harrison transformation on Schwarzschild produces the Reissner-Nordström metric whose outer horizon radius explicitly depends on Q. The manuscript must exhibit the explicit metric components (or at least the norm of the static Killing vector) in this limit and demonstrate either that the generated line element differs from RN or that a non-standard form of the transformation is being used that still satisfies the Einstein-Maxwell equations.

    Authors: We acknowledge the importance of this consistency check. We will revise the manuscript to include the explicit metric components (and the norm of the static Killing vector) obtained by setting the distortion parameter to zero. This addition will demonstrate the precise form of the generated line element in the limit, confirm that it satisfies the Einstein-Maxwell equations, and either establish that a non-standard implementation of the Harrison transformation is employed or qualify the horizon-independence statement accordingly. The abstract and construction section will be updated to reflect the clarified properties. revision: yes

  2. Referee: [Thermodynamics] Thermodynamics section: if the horizon radius is truly independent of Q, the first law and Smarr relation must be re-derived with the electromagnetic potential appearing only as an external parameter; the manuscript should show the explicit form of these relations and verify consistency with the area law when the seed is Schwarzschild.

    Authors: We agree that explicit derivations are required. In the revised manuscript we will derive the first law and Smarr relation in detail, with the electromagnetic potential entering as an external parameter, and verify consistency with the area law for the Schwarzschild seed. These explicit forms will be added to the thermodynamics section. revision: yes

Circularity Check

0 steps flagged

No significant circularity; construction from external seed via standard transformation

full rationale

The paper constructs the electrically charged distorted black hole by applying the Harrison transformation to a static distorted vacuum seed metric, with the non-shift of the Killing horizon following directly from the stated preservation of radial structure under that transformation. This is a feature of the construction method itself rather than a derived claim that reduces to the result by definition. No load-bearing steps involve fitted parameters renamed as predictions, self-citation chains that substitute for independent justification, or ansatze smuggled via prior work by the same authors. Thermodynamic quantities, particle orbits, shadow, and quasinormal modes are computed from the resulting metric without equations that equate outputs to inputs by construction. The derivation remains self-contained against the external seed and the known properties of the Harrison transformation.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The construction rests on the Einstein-Maxwell equations and the existence of a suitable static distorted vacuum seed; the distortion parameter is introduced to deform the geometry but is not fitted to data in the abstract.

free parameters (1)
  • distortion parameter
    Controls the deviation from spherical symmetry in the seed metric.
axioms (2)
  • standard math Einstein-Maxwell field equations
    The underlying theory in which the solution is constructed.
  • domain assumption Existence of static distorted vacuum seed metric amenable to Harrison transformation
    Required starting point for generating the charged solution while preserving radial structure.

pith-pipeline@v0.9.1-grok · 5807 in / 1265 out tokens · 34037 ms · 2026-06-26T01:38:52.431583+00:00 · methodology

discussion (0)

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